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Part Exploring and Understanding Data Chapter 5 Describing Distributions Numerically 45
10. Standard deviation.
a) Set 2 has the greater standard deviation. Both sets have the same mean (7), maximum (10), 20. Gas prices. and minimum (4), but 6 and 8 are farther from the mean than 7. a) SD(Set 1) =have been increasing 2.24average over the three year period, and the spread has Gas prices 2.12 SD(Set 2) = on been increasing as well. The distribution of prices in 2002 was skewed to the left with b) The standard deviations are the same. Set 1 is simply Set 2 + 90. Although the measures of several low outliers. Since then, the distribution has been increasingly skewed to the right. center and position are different, the spread is exactly the same. There is a high outlier in 2004, although it appears to be pretty close to the upper fence. SD(Set 1) = 36.06 SD(Set 2) = 36.06 b) The distribution of gas prices in 2004 shows the greatest range and the biggest IQR, so the c) Set 2 has the greater standard deviation. The central 4 values of Set 2 are simply the central prices varied a great deal. 4 values of Set 1 +40, but the maximum and minimum of Set 2 are farther away from the 21. Marriagethan the maximum and minimum of Set 1. Range(Set 1) = 18 and Range(Set 2) = 22. mean age. Since the Range of Set 2 is greater than the Range of Set right, with a typical man (asalso The distribution of marriage age of U.S. men is skewed 1, the standard deviation is larger. measured by the median) first marrying 44 Part I Exploring and Understanding Data at around 24 years old. The middle 50% of male SD(Set 1) ages is between about 7.04 26 years. For U.S. women, the distribution of SD(Set 2) = 23 and marriage = 6.03
marriage 11. States.sites. age is also skewed right, with median of around 21 years. The middle 50% of 16. Camp female marriage age is between about 20 and 23 years. When comparing the two a) The distribution of state populations is skewed public parks inright. Therefore, the to the heavily to the median a) The distribution of the number feature is that the distributions Vermont isidentical in distributions, the most striking of campsites in are nearly skewed and IQR medianappropriate measures of center and of center and spread. are the right, sobut have and IQR are appropriate measures spread. marry earlier than males. In spread, different centers. Females typically seem to b) The mean Q1 = 78 28 = 50. the women marry at a younger age than any man.values on fact, = Q3 population must of larger than the median population. The extreme between 50% and 75% be b) IQR the right affect theRule (1.5 IQRs and have no effect on the median. Using the Outlier mean greatly beyond quartiles): 22. Fuel economy. c) There are 51 entries in the stemplot, so the 26th entry must be the median. Counting in the Upper Fence: Q3 + 1.5(IQR) get + 1.5(50) Cars with 4 cylinders generally= 78 better gas mileage than cars with 6 cylinders, which ordered stemplot gives median = 4 million people. The middle of the lower 50% of the list generally get better gas mileage than 75th with 8th cylinders. Additionally, the greater the = the cars (26 state populations) is between 78 + 13 and 14 population, or 1.5 million people. The number of cylinders, the more consistent the mileage becomes. 4 cylinder cars typically get middle of the upper half of the = 153 state populations) is between the 13th and 14th list (26 between 27 33 mpg, 6 cylinder cars typically get between 18 22 mpg, and large cars population from the top, or 6 million people. The IQR = Q3 Q1 = 6 1.5 = 4.5 million Lower Fence:between 16 0 campsites. typically get Well below 19 mpg. people. There 23. Grapes. are 3 parks with greater than 180 campsites. These are definitely outliers. There are d) The distribution of population160 campsites each. These may 2 parks with between 150 and for the 50 U.S. States and Washington, D.C. is skewed a) heavily toLakeright. The had the maximum case price of people, 50% with of states having A outliers as vineyard median population is 4 million approximately $150. beSeneca the well. 300 b) populations between 1 and 6 the minimumcampsites is at the A boxplotLake vineyard had of number of case price of approximately $50. 34 million Seneca of the distribution million people. There is one outlier, a state with c) A people. The next highest population is only 21 million. 225 c) right. The Keuka Lake vineyards were consistently high. 12. Wayne Gretzky. 150 d) The distribution of the number of campsites at public parks in Cayuga Lake vineyards and Seneca Lake vineyards have approximately the same average a) The distribution of the number a typical Keuka The center of has a case price of about to Vermont isat about $100, while of games right. Lakeseason by Wayne Gretzky is skewed case price, unimodal and skewed to the played per vineyard 75 the left,Keuka Lake vineyards The44consistently The case prices, between $120 and $140, the distribution is approximately median is more resistant to the skewness and outliers $130. and has low outliers. have campsites. high than the mean. campsites is quite spread out, with severalvineyards have case prices from 0 distribution of with one low outlier at about $85 per case. Cayuga Lake high outliers. These middle of the ordered list, campsites each. case prices, 11th $50 to $150. $70 median, or Seneca Lake excess of have highly variable sites b) The to $135, andparks have in vineyards150is 79 games. Both the 10th and fromvalues are 79, 17. Sipso the median is the average of these two, also 79. size. 24. Ozone. c) The mean the highest recorded ozone level, approximately 440. noplayed an That makes it a) The distribution of height is Therenearly symmetric and shows April had should be lower. most are two seasons when Gretzky outliers. unusually low number candidate for summarizing with a mean. The distribution of weight is skewed to the best of games. Those seasons will pull the mean down. b) February had the largest IQR of ozone level, approximately 50. the right, which would inflate the value of the mean. The distribution of sip size has an c) outlier, which would inflate the value oflevels, approximately 50. August had the smallest range of ozone the mean. d) January had a slightly size shows a high outlier. The June, 340 and 350, is sensitive but b) The distribution of sip lower median ...